The research of the mapping category workforce Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and workforce idea. This booklet explains as many vital theorems, examples, and strategies as attainable, fast and at once, whereas whilst giving complete info and protecting the textual content approximately self-contained. The ebook is acceptable for graduate students.A Primer on Mapping classification teams starts via explaining the most group-theoretical houses of Mod(S), from finite new release through Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the best way, primary items and instruments are brought, reminiscent of the Birman specified series, the complicated of curves, the braid workforce, the symplectic illustration, and the Torelli staff. The ebook then introduces Teichmller area and its geometry, and makes use of the motion of Mod(S) on it to turn out the Nielsen-Thurston class of floor homeomorphisms. subject matters contain the topology of the moduli area of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov conception, and Thurston's method of the type.

In accordance with lectures given at a tutorial direction, this quantity permits readers with a uncomplicated wisdom of practical research to entry key study within the box. The authors survey a number of components of present curiosity, making this quantity perfect preparatory examining for college kids embarking on graduate paintings in addition to for mathematicians operating in comparable components.

"The textual content can function an creation to basics within the respective components from a residuated-maps standpoint and with a watch on coordinatization. The old notes which are interspersed also are worthy declaring. …The exposition is thorough and all proofs that the reviewer checked have been hugely polished.

The normal biennial foreign convention of abelian workforce theorists was once held in August, 1987 on the college of Western Australia in Perth. With a few forty members from 5 continents, the convention yielded various papers indicating the fit nation of the sphere and exhibiting the major advances made in lots of components because the final such convention in Oberwolfach in 1985.

13 gives us a way to replace homeomorphisms with diffeomorphisms. We can also replace isotopies with smooth isotopies. In other words, if two diffeomorphisms are isotopic, then they are smoothly isotopic; see, for example, [30]. In this book, we will switch between the topological setting and the smooth setting as is convenient. For example, when deﬁning a map of a surface to itself (either by equations or by pictures), it is often easier to write down a homeomorphism than a smooth map. On the other hand, when we need to appeal to transversality, extension of isotopy, and so on, we will need to assume we have a diffeomorphism.

Nonseparating simple proper arcs in a surface S that meet the same number of components of ∂S. 6. Chains of simple closed curves. A chain of simple closed curves in a surface S is a sequence α1 , . . , αk with the properties that i(αi , αi+1 ) = 1 for each i and i(αi , αj ) = 0 whenever |i − j| > 1. A chain is nonseparating if the union of the curves does not separate the surface. Any two nonseparating chains of simple closed curves with the same number of curves are topologically equivalent. This can be proved by induction.

They correspond to elements of PSL(2, R) whose trace has absolute value less than 2. Parabolic. If f has exactly one ﬁxed point in ∂H2 , then f is called parabolic. In the upper half-plane model, f is conjugate in Isom+ (H2 ) to z → z ± 1. Parabolic isometries correspond to those nonidentity elements of PSL(2, R) with trace ±2. Hyperbolic. If f has two ﬁxed points in ∂H2 , then f is called hyperbolic or loxodromic. In this case, there is an f -invariant geodesic axis γ; that is, an f -invariant geodesic in H2 on which f acts by translation.